Hehe, no I didn't. I was just googling at Fibonacci numbers or something like that and happened to find it. Then I started to think about how to do to show it's true, but I realized I don't know.

Another thing you can gather from this is that the "diagonal" sum is growing with an approximate factor of , the golden ratio, while the horozontal sum in growing with a factor of 2. So when you take steps withe the diagonal, the factor of growing is , while it would require only steps for the sum to grow as much if the sum would have been taking from a horitzonal line, where . So which means it requires times as many steps to make the diagonal sum grow with as big factor as the horitzonal sum does.

Let me explain it,
The Binet formula (I derived it on the forum somewhere) is,
Where,
And,
Thus, for large we have, because .
Thus,

If that's your definition for then I suggest you use instead:

Btw I think you use to have for the small value. Then I'm not sure whether the definition is or if it is . If it is the second, then both values can be derived from the equation , cause then you will get the answer . On the other hand, if it is the first definition then both values can be given by the equation . Hm, I got the equation can be derived from the equation , or , I wonder where that equation comes from...